Note on beta elements in homotopy, and an application to the prime three case

Let $S^0_{(p)}$ denote the sphere spectrum localized at an odd prime $p$. Then we have the first beta element $\beta_1\in\pi_{2p^2-2p-2}(S^0_{(p)})$, whose cofiber we denote by $W$. We also consider a generalized Smith-Toda spectrum $V_r$ characterized by $BP_*(V_r)=BP_*/(p,v_1^r)$. In this note, we show that an element of $\pi_*(V_r\wedge W)$ gives rise to a beta element of homotopy groups of spheres. As an application, we show the existence of $\beta_{9t+3}$ at the prime three to complete a conjecture of Ravenel's: $\beta_{s}\in \pi_{16s-6}(S^0_{(3)})$ exists if and only if $s$ is not congruent to $4$, $7$ or $8$ mod $9$.